Black artifacts on plane with procedural displacement
Sorry for the late answer, but those are not artifacts. The person in the tutorial used procedural displacement to create "waves" in the water by adding a noise texture to displace the water surface, then stretching it in one direction. If you look closely, the black marks are actually parts of the waves.

Take a look at the noise texture you used to create this effect and make sure it is scaled properly. This also makes sense because displacing a more high-res geometry just results in more intricate details like you have noticed.

A little advice for later on: when following a tutorial, don't just try to copy everything. That will get you a result, but in the end, you have learned nothing. Try to understand what the person is doing. This will teach you a lot, and will also make it easier to find and solve mistakes.